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Victoria city and Sendai city

Victoria city and Sendai city. 7300km. Victoria city. Sendai city. Tohoku University. Tohoku University was established in 1907. Spring. Summer. Autumu. Winter. GSIS, Tohoku University. Graduate School of Information Sciences ( GSIS ), Tohoku University, was established in 1993.

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Victoria city and Sendai city

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  1. Victoria city and Sendai city 7300km Victoria city Sendai city

  2. Tohoku University Tohoku University was established in 1907. Spring Summer Autumu Winter

  3. GSIS, Tohoku University Graduate School of Information Sciences (GSIS), Tohoku University, was established in 1993. 150 Faculties 450 students Math. Computer Science Robotics Transportation Economics Human Social Sciences Interdisciplinary School

  4. Book

  5. Small Grid Drawings of Planar Graphs with Balanced Bipartition Xiao ZhouTakashi HikinoTakao Nishizeki Graduate School of Information Sciences, Tohoku University, Japan

  6. Grid drawing In a grid drawing of a planar graph, 　　・ every vertex is located at a grid point, 　　・ every edge is drawn as a straight-line segment without any edge-intersection. Planar graph Grid drawing 2 2 3 3 4 4 1 1 5 5 6 6 7 7

  7. Embedding We deal with grid drawings of a planar graph in variable embedding setting. 1 2 3 4 5 6 This embedding is different from a given embedding 7 Planar graph Grid drawing 2 2 3 3 4 4 1 1 5 5 6 6 7 7

  8. Width and Height of grid drawing H H W W W=9, H=11 W=4, H=3 Area W×H=99 Area W×H=12

  9. Small grid drawing Large area Small area H H W W We wish to find a small grid drawing in variable embedding setting.

  10. Known results n ： number of vertices

  11. u1 u2 Our results If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. s s Drawing Bipartition G2 s G2 s G2 G1 t G1 G1 t G t t Planar graph G Subgraph G1,G2 Drawing of G

  12. u1 u1 u1 u2 u2 u2 Outline of algorithm Maximal planar graph s Bipartition s G2 s G2 t G1 G1 s t G G2 t s t (1) Planar graph G (2) Subgraph G1,G2 G1 t (3) Maximal planar graph G1,G2 Drawing s Combining s s G2 G2 G1 G1 t t t (5) Drawing of G (4) Drawing of G1,G2

  13. u1 u2 Our results s G2 Theorem 1 G1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. t G (1) Planar graph G W≤max{n1, n2}－1 s W,H≤2n/3 n1,n2≤2n/3 G2 H≤max{n1, n2}－2 G1 W,H≤αn n1,n2≤αn t If α<1, Balanced bipartition (5) Drawing of G

  14. Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. W,H≤2n/3 n1,n2≤2n/3 W,H≤αn n1,n2≤αn If α<1, Balanced bipartition

  15. Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallelgraph has a balanced bipartition (n1,n2≤2n/3). Planar graph α=2/3 Series-Parallel graph

  16. Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallelgraph has a balanced bipartition (n1,n2≤2n/3). Planar graph α=2/3 Series-Parallel graph Theorem 2 Series-Parallel graph W H

  17. u1 u1 u1 u2 u2 u2 Outline of algorithm Maximal planar graph s Bipartition s G2 s G2 t G1 G1 s t G G2 t s t (1) Planar graph G (2) Subgraph G1,G2 G1 t (3) Maximal planar graph G1,G2 Drawing s Combining s s G2 G2 G1 G1 t t t (5) Drawing of G (4) Drawing of G1,G2

  18. Bipartition We call a pair of distinct vertices {s,t} in a graph G=(V,E) a separation pair of G if G has two subgraphs G1=(V1,E1) and G2=(V2,E2) such that 　・ V=V1∪V2，V1∩V2={s,t}, 　・ E=E1∪E2，E1∩E2=∅． Such a pair of subgraphs {G1,G2} is called a bipartition of G. G2 s s s Bipartition n1=9 t t t G1 n2=5 n=12 (1) Graph G (2) Subgraph G1,G2

  19. Bipartition We call a pair of distinct vertices {s,t} in a graph G=(V,E) a separation pair of G if G has two subgraphs G1=(V1,E1) and G2=(V2,E2) such that 　・ V=V1∪V2，V1∩V2={s,t}, 　・ E=E1∪E2，E1∩E2=∅． Such a pair of subgraphs {G1,G2} is called a bipartition of G. G2 t t t G1 Bipartition s s s n1=3 n=12 n2=11 (1) Graph G (2) Subgraph G1,G2

  20. u1 u1 u1 u2 u2 u2 Outline of algorithm Maximal planar graph s Bipartition s G2 s G2 t G1 G1 s t G G2 t s t (1) Planar graph G (2) Subgraph G1,G2 G1 t (3) Maximal planar graph G1,G2 Drawing s Combining s s G2 G2 G1 G1 t t t (5) Drawing of G (4) Drawing of G1,G2

  21. u1 u2 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. W≤max{n1, n2}－1 Drawing in linear time s s G2 G2 H≤max{n1, n2}－2 G1 G1 t G t Planar graph G Drawing of G

  22. Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Assume w.l.o.g. that n1≥n2． s s s G1 t t t G2 n2=5 G1 n1=9 G n=12

  23. Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Assume w.l.o.g. that n1≥n2． Add dummy edges to G1 so that the resulting graph is maximal planar and has an edge (s,t). s s G1 t t G1 n1=9 G n=12

  24. u1 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Embed G1 so that the edge (s,t) lies on the outer face of G1. s s G1 t t G1 n1=9 G n=12

  25. u1 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Obtain a grid drawing of G1[CK97]. s s G1 t t G1 n1=9 G n=12

  26. u1 u1 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. s Obtain a grid drawing of G1[CK97]. t G1 s n1=9 H1=n1－2 Edge (u1,t) is horizontal. G1 n1=9 t W1=n1－2

  27. u1 u2 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. s G2 s s G2 t t G2 n2=5 t G n=12 G1 n1=9

  28. u1 u2 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Add n1－n2 dummy vertices to G2 so that the resulting graph has exactly n1 vertices. s G2 s s G2 t t G2 n2=5 n2=n1=9 t G n=12 G1 n1=9

  29. Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Add dummy edges to G2 so that the resulting graph is maximal planar and has an edge (s,t). G2 s s t t G2 n2=n1=9 G n=12

  30. Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Embed G2 so that the edge (s,t) lies on the outer face of G2. G2 s s t t G2 n2=n1=9 G n=12

  31. u2 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Embed G2 so that the edge (s,t) lies on the outer face of G2. G2 s s s t t t G2 G2 n2=n1=9 n2=n1=9 G n=12

  32. t u2 u2 s Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Obtain a grid drawing of G2[CK97]. Edge (u2,s) is horizontal. G2 s s t t G2 n2=n1=9 G n=12

  33. u1 u2 Theorem 1 s s t t G1 G2 n1=9 n2=9 u2 s s u1 t t

  34. s u1 u1 u2 u2 t Theorem 1 G s Combine the two drawings and Erase all the dummy vertices and edges. t n=12 s G2 n2=9 G1 n1=9 t

  35. u1 u1 u2 u2 Theorem 1 G s Combine the two drawings and Erase all the dummy vertices and edges. t n=12 s G2 n2=9 G1 n1=9 t

  36. u1 u1 u2 u2 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}－1, H≤max{n1,n2}－2 and such a drawing can be found in linear time. Such a drawing can be found in linear time, because drawings of G1,G2 can be found in linear time by the algorithm in CK97. W2=n1－2 n1≥n2 s s G2 H=H1 =n1－2 =max{n1, n2}－2 G2 G1 H2=n1－2 s t t H1=n1－2 G1 t W=W1+1 =n1－1 =max{n1, n2}－1 Q.E.D. W1=n1－2

  37. Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallelgraph has a balanced bipartition (n1,n2≤2n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel graph W H

  38. Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallelgraph has a balanced bipartition (n1,n2≤2n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel graph W H

  39. Series-Parallel graph A Series-Parallelgraph is recursively defined as follows: (1) terminal A single edge is a SP graph． (2) (2) : SP graph G2 G2 G1 G1 Series connection SP graph G1 G2 G1 Parallel connection SP graph G2

  40. Series-Parallel graphs These graphs are Series-Parallel. s t

  41. Bipartition of Series-Parallel graph Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 . Furthermore such a bipartition can be found in linear time. Bipartition in liner time s s s G2 G2 t G1 G1 t t G SP graph G Subgraph G1,G2 n1,n2 Suppose for a contradiction that a SP graph has no desired bipartition.

  42. Bipartition of Series-Parallel graph Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 . Furthermore such a bipartition can be found in linear time. Let {s,t} be themost balanced separation pair ofG: max{n1,n2} is minimumamong all bipartitions of G. n1>2n/3 Assume w.l.o.g. that n1≥n2． G1=G11・G12 G1 G11 G11 u G12 G12 s t G2 n2<n/3 SP graph G 2-connected

  43. Bipartition of Series-Parallel graph Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 . Furthermore such a bipartition can be found in linear time. Let {s,t} be themost balanced separation pair ofG: max{n1,n2} is minimumamong all bipartitions of G. n1>2n/3 Assume w.l.o.g. that n1≥n2and n11≥n12. G1=G11・G12 n11>n/3 G1 G11 n1> n11 u G12 n1> n11 G11 s t n1> max{n11,n11} G2 Contradiction. n11<2n/3 n1=max{n1,n2} n2<n/3 max{n1,n2}> max{n11,n11} SP graph G 2-connected

  44. Grid drawing of Series-Parallel graph s Lemma 1 in linear time s s G2 G2 G1 G1 t t t G SP graph G Subgraph G1,G2 n1,n2

  45. u1 u2 Grid drawing of Series-Parallel graph s Lemma 1 in linear time s s G2 G2 G1 G1 t t t G SP graph G Subgraph G1,G2 n1,n2 s s Theorem 1 in linear time G2 s G2 H≤max{n1, n2}－2 G1 G1 t t t Subgraph G1,G2 W≤max{n1, n2}－1

  46. u1 u2 Grid drawing of Series-Parallel graph s Lemma 1 in linear time s s G2 G2 G1 G1 t t t G SP graph G Subgraph G1,G2 n1,n2 s s Theorem 1 in linear time G2 s G2 H≤max{n1, n2}－2 G1 G1 t t t SP Subgraph G1,G2 W≤max{n1, n2}－1 n1,n2

  47. u1 u2 Grid drawing of Series-Parallel graph Theorem 2. Every Series-Parallel graph of n vertices has a grid drawing such that WH . Furthermore such a drawing can be found in linear time. s s Theorem 1 in linear time G2 s G2 H≤max{n1, n2}－2 G1 G1 t t t SP Subgraph G1,G2 W≤max{n1, n2}－1 n1,n2

  48. u1 u2 Grid drawing of Series-Parallel graph s s Theorem 2 in linear time H=7 t SP graph G n=12 t W=8 s Lemma 1 in linear time Theorem 1 in linear time s G1 n1=9 G2 t t n2=5

  49. u1 u1 u2 u2 Conclusions W≤max{n1, n2}－1 s G2 H≤max{n1, n2}－2 G1 t W s G2 G1 H t

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